Lecture 2

1.2 Propositional Equivalences

Compound Propositions

Compound propositions are made by combining existing propositions using logical operators

Logical Equivalences

Example - DeMorgans Laws

When the number of variables are small, a truth table is an effective method of proof in which all possible truth value permutations are evaluated.

How many rows are in a truth table that enumerates an compound proposition that is comprised of N variables?

More Examples

Important Equivalences

Logical Equivalences Involving Quantifiers

Statements involving predicates and quantifiers are logically equivalent iff they have the same

truth value for all applications and for all domains of discourse.

)()()()( xQxxPxxQxPx

)()( xPxxxP

)()( xPxxxP

)()()()( xQxPxxQxPx

Negation of a Proposition

Let p be a proposition. The negation of p, denoted by ~p is the statement

"It is not the case that p."

or

"The proposition, p is false"

or

"Not p is true"

The proposition ~p is read "not p". The truth value of the negation of p, ~p, is the opposite of the truth value of p.

Pigs have wings.

I have a million dollars.

The negation operation will become more interesting when we introduce univeral and existential quantifiers.

Conditional Statements

Let p and q be propositions. The conditional statement (or implication) is the proposition "if p, then q." The conditional statement is false when p is true and q is false, and true otherwise. In this conditional statement p is the hypothesis (also called the antecedent or premise) and q is called the conclusion (also called the consequence).

qp

http://www.stanford.edu/class/cs103a/handouts/17%20Conditionals.pdf

Equivalances with Conditionals and Biconditionals

Biconditional Statements

Let p and q be propositions. The biconditional statement is the proposition "p if and only if q." The biconditional statement is true when p and q have the same truth values, and is false otherwise.

qp

http://www.stanford.edu/class/cs103a/handouts/17%20Conditionals.pdf

The converse of the conditional statement is .

Converse

pq qp

p q p->q q->p

0 0 1 1

0 1 1 0

1 0 0 1

1 1 1 1

Inverse

The inverse of the conditional statement is .qp qp

p q p->q ~p->~q

0 0 1 1

0 1 1 0

1 0 0 1

1 1 1 1

Contrapositive

The inverse of the conditional statement is . The contrapositivealways has the same truth value as the original statement.

pq qp

p q p->q ~q->~p

0 0 1 1

0 1 1 1

1 0 0 0

1 1 1 1

Testing System SpecificationsDetermine whether these system specifications are consistent:

"The diagnostic message is stored in the buffer or it is retransmitted"

"The diagnostic message is not stored in the buffer."

"If the diagnostic message is stored in the buffer, then it is retransmitted."

p = "The diagnostic message is stored in the buffer"q = "The diagnostic message is retransmitted."

a.

b.

c.

a.

b.

c. qp

p

qp

For the system to be consistent there must be at least one truth assignment for the variables that make all the statements true.

Add the specification, "The diagnostic message is not retransmitted." and determine if thesystem remains consistent.

a.

b.

c.

d. q

qp

p

qp

p q pvq ~p ~q p->q

0 0 0 1 1 1

0 1 1 1 0 1

1 0 1 0 1 0

1 1 1 0 0 1

Dual

The dual of a compound proposition that contains only the logical operators , , and is the compound proposition obtained by replacing each by , and each by , each T by F, and each F by T. The dual of s is denoted by s*.

Fqpqp Tqpqp

expression dual

Satisfiability

A compund proposition is satisfiable if there is an assignment of truth values to the variables in the compund proposition that makes the statement form true.

)()()()()( rqpqrqrppqp

0 0 0 0 1 1 1 1 1

0 0 1 0 1 0 1 1 1

0 1 0 1 1 1 0 0 1

0 1 1 1 1 0 1 0 0

1 0 0 1 0 1 1 1 1

1 0 1 1 0 1 1 1 1

1 1 0 1 0 1 0 0 1

1 1 1 1 0 1 1 0 1

rqrprqp

Problem:Let X and Y be straight lines and define,

p = "X and Y lie in the same plane" q = "X and Y never cross" r = "X and Y are parallel"

(1) verify that the logical expression is a tautology(2) rewrite the tautology as a statement using the propositions(3) explain the apparent inconsistency

The expression below is one of the logical equivalences listed in Table 6 of the textbook.

Puzzle